Energy of zeros of random sections on Riemann Surface

Families of Line Bundles over Riemann Surfaces, their Sections, and their Degenerations -- A Constructive Approach using Automorphic Forms

In this paper we study the deformation problem of pairs consisting of a Riemann surface and a holomorphic line bundle over that surface, and also sections thereof. We emphasize a constructive approach throughout and work and use covering space techniques. In particular, we also describe the limits of such degenerations as the boundary of Teichmüller space is approached, and review the construction of augmented Teichmüller space in great detail.

On the Riemann surface type of Random Planar Maps

We show that the (random) Riemann surfaces of the Angel-Schramm Uniform Infinite Planar Triangulation and of Sheffield's infinite necklace construction are both parabolic. In other words, Brownian motion on these surfaces is recurrent. We obtain this result as a corollary to a more general theorem on subsequential distributional limits of random unbiased disc triangulations, following work of Benjamini and Schramm.

Scaled Correlations of Critical Points of Random Sections on Riemann Surfaces

In this paper we prove that as N goes to infinity, the scaling limit of the correlation between critical points z1 and z2 of random holomorphic sections of the N-th power of a positive line bundle over a compact Riemann surface tends to 2/(3pi^2) for small sqrt(N)|z1-z2|. The scaling limit is directly calculated using a general form of the Kac-Rice formula and formulas and theorems of Pavel Bleher, Bernard Shiffman, and Steve Zelditch.

Holomorphic sections of line bundles on the jet spaces of the Riemann sphere

Fix a point $t_0$ in the circle $S^1$. The space $J^k(t_0, \mathbb{P}^1)$ of $k$-jets at $t_0$ of $C^{\infty}$ maps from $S^1$ to the Riemann sphere $\mathbb{P}^1$ is a $k+1$ dimensional complex algebraic manifold. We identify a class of holomorphic sections of line bundles on $J^k(t_0, \mathbb{P}^1)$.

紧黎曼曲面上随机零点空洞概率的研究

Hole probabilities of random zeros on compact Riemann surfaces

本文研究紧黎曼曲面上随机全纯截面零点的空洞概率，建立了其收敛速度估计。

On the dimension of bundle-valued Bergman spaces on compact Riemann surfaces

Given a holomorphic vector bundle $E$ over a compact Riemann surface $M$, and an open set $D$ in $M$, we prove that the Bergman space of holomorphic sections of the restriction of $E$ to $D$ must either coincide with the space of global holomorphic sections of $E$, or be infinite dimensional. Moreover, we characterize the latter entirely in terms of potential-theoretic properties of $D$.

Conditional expectations of random holomorphic fields on Riemann surfaces

We study two conditional expectations: the expected density of critical points of Gaussian random holomorphic sections of powers of a positive holomorphic line bundle over Riemann surfaces given that the random sections vanish at a point and the expected density of zeros given that the random sections has a fixed critical point. The main result is that these two rescaling conditional expectations have universal limits as the power of the line bundle tends to infinity. We will see that the short distance behavior between critical points and the conditioning zero is neutral while there is a repu…

On Asymptotic Weil-Petersson Geometry of Teichmüller Space of Riemann Surfaces

In this paper, we study the asymptotic geometry of Teichmuller space of Riemann surfaces and give bounds on the Weil-Petersson sectional curvature of Teichmuller space, in terms of the length of the shortest geodesic on the surface. This will also imply that the sectional curvature is not pinched from above or below by any negative constant.

The Weil--Petersson geometry of the moduli space of Riemann surfaces

In [4], Z. Huang showed that in the thick part of the moduli space $\mathcal{M}_g$ of compact Riemann surfaces of genus $g$, the sectional curvature of the Weil--Petersson metric is bounded below by a constant depending on injectivity radius, but independent of the genus $g$. In this article, we prove this result by a different method. We also show that the same result holds for Ricci curvature. For the universal Teichmüller space equipped with Hilbert structure induced by Weil--Petersson metric, we prove that its sectional curvature is bounded below by a universal constant.

de Branges spaces on compact Riemann surfaces and a Beurling-Lax type theorem

Using the notion of commutative operator vessels, this work investigates de Branges-Rovnyak spaces whose elements are sections of a line bundle of multiplicative half-order differentials on a compact real Riemann surface. As a special case, we obtain a Beurling-Lax type theorem in the setting of the corresponding Hardy space on a finite bordered Riemann surface.

Quillen connection and the uniformization of Riemann surfaces

The Quillen connection on ${\mathcal L} \rightarrow {\mathcal M}_g$, where ${\mathcal L}^*$ is the Hodge line bundle over the moduli stack of smooth complex projective curves curves ${\mathcal M}_g$, $g \geq 5$, is uniquely determined by the condition that its curvature is the Weil--Petersson form on ${\mathcal M}_g$. The bundle of holomorphic connections on ${\mathcal L}$ has a unique holomorphic isomorphism with the bundle on ${\mathcal M}_g$ given by the moduli stack of projective structures. This isomorphism takes the $C^\infty$ section of the first bundle given by the Quillen connection o…

穿孔黎曼曲面上的半正线丛：Bergman核与随机零点

Semipositive line bundles on punctured Riemann surfaces: Bergman kernels and random zeros

本文研究了穿孔黎曼曲面上半正线丛的高张量幂的Bergman核展开及随机零点，证明了半经典极限下高斯全纯截面零点的等分布、大偏差、中心极限定理和数方差等结果。

A Hodge theoretic projective structure on Riemann surfaces

Given any compact Riemann surface $C$, there is a canonical meromorphic 2--form $\widehatη$ on $C\times C$, with pole of order two on the diagonal $Δ\, \subset\, C\times C$, constructed in \cite{cfg}. This meromorphic 2--form $\widehatη$ produces a canonical projective structure on $C$. On the other hand the uniformization theorem provides another canonical projective structure on any compact Riemann surface $C$. We prove that these two projective structures differ in general. This is done by comparing the $(0,1)$--component of the differential of the corresponding sections of the moduli space…

黎曼曲面的非阿贝尔方法第一部分：朗斯基几何

A Non-Abelian Approach to Riemann Surfaces Part I: Wronskian Geometry

针对黎曼曲面上的射影平坦全纯向量丛开展研究，通过为其分配Wronskian线丛、定义亚纯截面除法，将Abel恒等式解释为对应线丛的第一陈类。